1Research Center for System Theory and Applications, Chongqing University of Posts and Telecommunications, Chongqing 400065, China2Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Abstract

A hybrid projective synchronization scheme for two identical fractional-order chaotic systems is proposed in this paper. Based on the stability theory of fractional-order systems, a controller for the synchronization of two identical fractional-order chaotic systems is designed. This synchronization scheme needs not to absorb all the nonlinear terms of response system. Hybrid projective synchronization for the fractional-order Chen chaotic system and hybrid projective synchronization for the fractional-order hyperchaotic Lu system are used to demonstrate the validity and feasibility of the proposed scheme.

1. Introduction

Most recently, many authors begin to investigate the chaotic dynamics and synchronization for fractional-order dynamical systems [1–6]. Chaos synchronization of the fractional-order systems is just beginning to attract some attention due to its potential applications in secure communications and control processing [7–12]. Several types of chaos synchronization are well known, which include complete synchronization (CS), antisynchronization (AS), phase synchronization, generalized synchronization (GS), projective synchronization (PS), and modified projective synchronization (MPS). Among all patterns of synchronization, the most noticeable one may be projective synchronization (PS), which was first studied by Mainieri and Rehacek [13]. Projective synchronization (PS) has been extensively considered because it can obtain faster communication. The drive and response system could be synchronized up to a scaling factor in projective synchronization. In application to secure communications, this proportional feature can be used to extend binary digital to M-nary digital communication for getting faster communication [14, 15].

However, most of projective synchronizations for the fractional-order systems have concentrated on studying the same scaling factor [16–19], and some projective synchronization schemes [16, 17] are suitable for a class of fractional-order systems or for some special fractional-order systems [19], and all the nonlinear terms of response system was absorbed in some previous works. Moreover, in order to increase the degree of secrecy for secure communications, the same scaling factor in PS can be replaced by vector function factor. Motivated by the above discussions, we propose a hybrid projective synchronization (HPS) scheme for two identical fractional-order chaotic systems in this paper. Hybrid projective synchronization (HPS) is a more general definition of projective synchronization, in which the drive system and response system could be synchronized up to a vector function factor. HPS is different from the PS. Furthermore, HPS could be used to get more secure communication than PS in application to secure communications, because it is obvious that the unpredictability of the vector function factor in HPS is more than that of the same scaling factor in PS. The main contribution of this paper is as follows: the HPS scheme in this paper is suitable for a large number of fractional-order systems (not for special fractional-order system), and this HPS scheme needs not to absorb all the nonlinear terms of response system. This is different from some previous works [16, 17, 19–21].

To illustrate the effectiveness of the proposed HPS scheme in this paper, the HPS for the fractional-order Chen system and HPS for the fractional-order hyperchaotic Lu system are investigated. Numerical simulations are used to demonstrate the effectiveness of the proposed schemes. The organization of this paper is as follows. In Section 2, the definition of HPS is given, and a HPS scheme for two identical fractional-order chaotic systems is presented. In Section 3, two groups of examples are used to verify the effectiveness of the proposed scheme. The conclusion is finally drawn in Section 4.

2. Hybrid Projective Synchronization Scheme

There are several definitions of fractional derivatives. In this paper, the Caputo-type fractional derivative defined will be used. The Caputo definition of the fractional derivative, which is sometimes called smooth fractional derivative, is described as
where is fractional order and denote the Caputo definition of the fractional derivative. is the smallest integer larger than , and is the -order derivative in the usual sense. is the gamma function.

The fractional-order chaotic drive and response systems can be written as follows, respectively:
where , are state vectors of the drive system (2.2) and the response system (2.3) and are continuous vector functions, respectively. is a vector controller.

Definition 2.1. For the drive system (2.2) and the response system (2.3), it is said to be hybrid projective synchronization (HPS) if there exists an reversible matrix such that
where is the Euclidean norm.

Remark 2.2. If , and is a unit matrix, then this synchronization is called complete synchronization (CS); if , then this synchronization is called antisynchronization (AS); if , and is a nonzero real constant, then this synchronization is called projective synchronization (PS); if , and are not the same nonzero constant, then this synchronization is called modified projective synchronization (MPS). Therefore, CS, AS, PS, and MPS are the special cases of the hybrid projective synchronization scheme (HPS) in this paper.

In order to realize HPS for the fractional-order chaotic system (2.2), we take the fractional-order chaotic system (2.2) as drive system and construct a response system as follows:
where is the reverse matrix of the reversible matrix , are state vector of the response system (2.5), and is a controller which will be designed.

Define the HPS errors between the response system (2.5) and the drive system (2.2) as
where

Let

Now, we assume that the errors vector can be divided into and , such that has the form of
where , , and , respectively. and are real constant matrix.

Rewrite the controller as follows:
where and , respectively.

Now, Theorem 2.3 is given based on the previously mentioned conditions in order to achieve the HPS between the drive system (2.2) and the response system (2.5).

Theorem 2.3. Choose the following controller:
where and are suitable constant matrix, respectively.

If all the eigenvalues of satisfy () and all the eigenvalues of satisfy (), then hybrid projective synchronization between the drive system (2.2) and the response system (2.5) can be achieved.

Proof. According to the drive system (2.2) and the response system (2.5), the error dynamic system of hybrid projective synchronization can be obtained as follows:
According to (2.9) and (2.10), the error dynamic system (2.12) can be rewritten as
because
So,
Because all the eigenvalues of satisfy (), according to the stability theory of linear fractional-order systems [22], the equilibrium point in the first equation of system (2.15) is asymptotically stable, which indicates . Since and , therefore when time , the second equation of system (2.15) can be changed as
because all the eigenvalues of satisfy (). According to the stability theory of linear fractional-order systems [22], the equilibrium point of system (2.16) is asymptotically stable, which indicates . According to and , the hybrid projective synchronization between the drive system (2.2) and the response system (2.5) can be achieved. This finishes the proof.

Remark 2.4. In order to use the stability theory of linear fractional-order systems [22], the controller or are chosen as . Moreover, the nonlinear term in the error dynamic system (2.13) or response system (2.5) is preserved; this is different from some previous works [17, 19–21] which need to absorb all the nonlinear terms of response system or error dynamic system.

Remark 2.5. For the complex fractional-order multiscroll chaotic systems [23–25] and the complex dynamical network or the small-world dynamical networks [26–28], the hybrid projective synchronization would be much more complex. Further work on this issue is an ongoing research topic in our group.

3. Applications

In order to illustrate the effectiveness of the proposed hybrid projective synchronization scheme obtained in Section 2, two examples are considered in this section, which are HPS for the fractional-order Chen system and HPS for the fractional-order hyperchaotic Lu system.

According to Theorem 2.3 in Section 2, the controller can be chosen as follows:
where , , , and , respectively. Therefore, choose suitable matrix and . If the eigenvalues of satisfy and if all the eigenvalues of satisfy (), then hybrid projective synchronization between the drive system (3.1) and its response system (3.2) can be achieved.

For example, choose reversible matrix

So, the eigenvalues of are , and all the eigenvalues of are and (), respectively. Therefore, the hybrid projective synchronization between drive system (3.1) and its response system (3.2) can be achieved. The corresponding numerical result is shown in Figure 2, in which the initial conditions are (, , ) = (3, 4, 5) for the drive system (3.1), and (, , ) = (8, 10, 4) for the response system (3.2), respectively, and .

Figure 2: The HPS result between the drive system (3.1) and its response system (3.2) for .

3.2. HPS for Fractional-Order Hyperchaotic Lǔ System

Min et al. reported a fractional-order hyperchaotic Lu system [30, 31] based on the hyperchaotic Lu system, which is described as follows:

The chaotic attractor of fractional-order hyperchaotic Lu system for , is shown in Figure 3.

Taking system (3.6) as the drive system, according to the HPS scheme presented in Section 2, the response system is described by
where is a reversible matrix and is its reverse matrix. As the same as what is mentioned previously, we can obtain

According to Theorem 2.3 in Section 2, the controller can be chosen as follows:
where , , and , respectively. Therefore, choose suitable matrix , and . If the eigenvalues of satisfy and if all the eigenvalues of satisfy (), then the hybrid projective synchronization between the drive system (3.6) and its response system (3.7) can be achieved.

For example, choose reversible matrix

So, the eigenvalues of are −36 and −10, and the eigenvalues of are −3 and −1, respectively. Therefore, the hybrid projective synchronization between drive system (3.6) and its response system (3.7) can be achieved. The corresponding numerical result is shown in Figure 4, in which the initial conditions are (, , ) = (3,4,5,6) for the drive system (3.6) and (, , , ) = (16,7,12,4) for the response system (3.7), respectively, and .

Figure 4: The HPS result between the drive system (3.6) and its response system (3.7) for .

4. Conclusions

We proposed a new synchronization scheme to achieve hybrid projective synchronization for two identical fractional-order chaotic systems in this paper. The drive system and response system could be synchronized up to a vector function factor, and this synchronization scheme needs not to absorb all the nonlinear terms of response system. The synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. Numerical simulations are used to illustrate the effectiveness of the proposed synchronization method.

Acknowledgments

The authors are very grateful to the reviewers for their valuable comments and suggestions, which have led to the improved presentation of this paper. This work is supported by Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ110525.